/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 418 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 3458 ms] (4) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: f0(A, B) -> Com_1(f2(A, B)) :|: A >= 1 && B >= 1 f0(A, B) -> Com_1(f2(A, B)) :|: A >= 1 && 0 >= B + 1 f0(A, B) -> Com_1(f2(A, B)) :|: 0 >= A + 1 && B >= 1 f0(A, B) -> Com_1(f2(A, B)) :|: 0 >= A + 1 && 0 >= B + 1 f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= B && B >= 2 && A >= 1 && A + B >= 2 f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= B && B >= 2 && 0 >= A + 1 && A + B >= 2 f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= B && 0 >= B && A >= 1 && A + B >= 2 f2(A, B) -> Com_1(f2(A, B - 1)) :|: A >= B && 0 >= B && 0 >= A + 1 && A + B >= 2 f2(A, B) -> Com_1(f2(A - 1, B)) :|: 1 >= A + B && A >= B + 1 && A >= 2 && B >= 1 f2(A, B) -> Com_1(f2(A - 1, B)) :|: 1 >= A + B && A >= B + 1 && A >= 2 && 0 >= B + 1 f2(A, B) -> Com_1(f2(A - 1, B)) :|: 1 >= A + B && A >= B + 1 && 0 >= A && B >= 1 f2(A, B) -> Com_1(f2(A - 1, B)) :|: 1 >= A + B && A >= B + 1 && 0 >= A && 0 >= B + 1 f2(A, B) -> Com_1(f2(A, B + 1)) :|: B >= A && B >= 0 && A >= 1 && 0 >= B + A + 1 f2(A, B) -> Com_1(f2(A, B + 1)) :|: B >= A && B >= 0 && 0 >= A + 1 && 0 >= B + A + 1 f2(A, B) -> Com_1(f2(A, B + 1)) :|: B >= A && 0 >= B + 2 && A >= 1 && 0 >= B + A + 1 f2(A, B) -> Com_1(f2(A, B + 1)) :|: B >= A && 0 >= B + 2 && 0 >= A + 1 && 0 >= B + A + 1 f2(A, B) -> Com_1(f2(A + 1, B)) :|: B + A >= 0 && B >= A + 1 && A >= 0 && B >= 1 f2(A, B) -> Com_1(f2(A + 1, B)) :|: B + A >= 0 && B >= A + 1 && A >= 0 && 0 >= B + 1 f2(A, B) -> Com_1(f2(A + 1, B)) :|: B + A >= 0 && B >= A + 1 && 0 >= A + 2 && B >= 1 f2(A, B) -> Com_1(f2(A + 1, B)) :|: B + A >= 0 && B >= A + 1 && 0 >= A + 2 && 0 >= B + 1 The start-symbols are:[f0_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 8*Ar_1 + 3*Ar_0^2 + 4*Ar_0*Ar_1 + 6*Ar_0 + Ar_1^2 + 4) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ 0 >= Ar_0 + 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ 0 >= Ar_0 + 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ Ar_0 >= 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ Ar_0 >= 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 1: f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ 0 >= Ar_0 + 1 /\ Ar_0 + Ar_1 >= 2 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ 0 >= Ar_0 + 1 /\ Ar_0 + Ar_1 >= 2 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ Ar_1 >= 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_1 >= 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ Ar_0 >= 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ Ar_0 >= 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ 0 >= Ar_1 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ 0 >= Ar_1 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 + V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] strictly and produces the following problem: 4: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] strictly and produces the following problem: 5: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: Ar_0*Ar_1 + Ar_1^2 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 + V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] strictly and produces the following problem: 6: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: Ar_0*Ar_1 + Ar_1^2 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = -V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] strictly and produces the following problem: 7: T: (Comp: 3*Ar_1 + Ar_0^2 + Ar_0*Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: Ar_0*Ar_1 + Ar_1^2 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] strictly and produces the following problem: 8: T: (Comp: 3*Ar_1 + Ar_0^2 + Ar_0*Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: Ar_0^2 + Ar_0*Ar_1 + Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: Ar_0*Ar_1 + Ar_1^2 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 - V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] strictly and produces the following problem: 9: T: (Comp: 3*Ar_1 + Ar_0^2 + Ar_0*Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: Ar_0^2 + Ar_0*Ar_1 + Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: Ar_0*Ar_1 + Ar_1^2 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] strictly and produces the following problem: 10: T: (Comp: 3*Ar_1 + Ar_0^2 + Ar_0*Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0^2 + Ar_0*Ar_1 + Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: Ar_0^2 + Ar_0*Ar_1 + Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: Ar_0*Ar_1 + Ar_1^2 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 - V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] strictly and produces the following problem: 11: T: (Comp: 3*Ar_1 + Ar_0^2 + Ar_0*Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0^2 + Ar_0*Ar_1 + Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: Ar_0^2 + Ar_0*Ar_1 + Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: Ar_0*Ar_1 + Ar_1^2 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 8*Ar_1 + 3*Ar_0^2 + 4*Ar_0*Ar_1 + 6*Ar_0 + Ar_1^2 + 4 Time: 0.470 sec (SMT: 0.387 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: f0 0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1 1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1 2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1 3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1 4: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 && A>=1 && A+B>=2 ], cost: 1 5: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 && 0>=1+A && A+B>=2 ], cost: 1 6: f2 -> f2 : B'=-1+B, [ A>=B && 0>=B && A>=1 && A+B>=2 ], cost: 1 7: f2 -> f2 : B'=-1+B, [ A>=B && 0>=B && 0>=1+A && A+B>=2 ], cost: 1 8: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 && B>=1 ], cost: 1 9: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 && 0>=1+B ], cost: 1 10: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && 0>=A && B>=1 ], cost: 1 11: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && 0>=A && 0>=1+B ], cost: 1 12: f2 -> f2 : B'=1+B, [ B>=A && B>=0 && A>=1 && 0>=1+A+B ], cost: 1 13: f2 -> f2 : B'=1+B, [ B>=A && B>=0 && 0>=1+A && 0>=1+A+B ], cost: 1 14: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B && A>=1 && 0>=1+A+B ], cost: 1 15: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B && 0>=1+A && 0>=1+A+B ], cost: 1 16: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && A>=0 && B>=1 ], cost: 1 17: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && A>=0 && 0>=1+B ], cost: 1 18: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A && B>=1 ], cost: 1 19: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A && 0>=1+B ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1 Removed rules with unsatisfiable guard: Start location: f0 0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1 1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1 2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1 3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1 4: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 && A>=1 && A+B>=2 ], cost: 1 6: f2 -> f2 : B'=-1+B, [ A>=B && 0>=B && A>=1 && A+B>=2 ], cost: 1 9: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 && 0>=1+B ], cost: 1 11: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && 0>=A && 0>=1+B ], cost: 1 13: f2 -> f2 : B'=1+B, [ B>=A && B>=0 && 0>=1+A && 0>=1+A+B ], cost: 1 15: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B && 0>=1+A && 0>=1+A+B ], cost: 1 16: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && A>=0 && B>=1 ], cost: 1 18: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A && B>=1 ], cost: 1 Simplified all rules, resulting in: Start location: f0 0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1 1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1 2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1 3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1 4: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 ], cost: 1 6: f2 -> f2 : B'=-1+B, [ 0>=B && A>=1 && A+B>=2 ], cost: 1 9: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 ], cost: 1 11: f2 -> f2 : A'=-1+A, [ A>=1+B && 0>=A ], cost: 1 13: f2 -> f2 : B'=1+B, [ B>=0 && 0>=1+A && 0>=1+A+B ], cost: 1 15: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B ], cost: 1 16: f2 -> f2 : A'=1+A, [ B>=1+A && A>=0 ], cost: 1 18: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A ], cost: 1 ### Simplification by acceleration and chaining ### Accelerating simple loops of location 1. Accelerating the following rules: 4: f2 -> f2 : B'=-1+B, [ A>=B && B>=2 ], cost: 1 6: f2 -> f2 : B'=-1+B, [ 0>=B && A>=1 && A+B>=2 ], cost: 1 9: f2 -> f2 : A'=-1+A, [ 1>=A+B && A>=1+B && A>=2 ], cost: 1 11: f2 -> f2 : A'=-1+A, [ A>=1+B && 0>=A ], cost: 1 13: f2 -> f2 : B'=1+B, [ B>=0 && 0>=1+A && 0>=1+A+B ], cost: 1 15: f2 -> f2 : B'=1+B, [ B>=A && 0>=2+B ], cost: 1 16: f2 -> f2 : A'=1+A, [ B>=1+A && A>=0 ], cost: 1 18: f2 -> f2 : A'=1+A, [ A+B>=0 && B>=1+A && 0>=2+A ], cost: 1 Accelerated rule 4 with metering function -1+B, yielding the new rule 20. Accelerated rule 6 with metering function -1+A+B, yielding the new rule 21. Accelerated rule 9 with metering function -1+A, yielding the new rule 22. Accelerated rule 11 with metering function A-B, yielding the new rule 23. Accelerated rule 13 with metering function -A-B, yielding the new rule 24. Accelerated rule 15 with metering function -1-B, yielding the new rule 25. Accelerated rule 16 with metering function -A+B, yielding the new rule 26. Accelerated rule 18 with metering function -1-A, yielding the new rule 27. Removing the simple loops: 4 6 9 11 13 15 16 18. Accelerated all simple loops using metering functions (where possible): Start location: f0 0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1 1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1 2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1 3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1 20: f2 -> f2 : B'=1, [ A>=B && B>=2 ], cost: -1+B 21: f2 -> f2 : B'=1-A, [ 0>=B && A>=1 && A+B>=2 ], cost: -1+A+B 22: f2 -> f2 : A'=1, [ 1>=A+B && A>=1+B && A>=2 ], cost: -1+A 23: f2 -> f2 : A'=B, [ A>=1+B && 0>=A ], cost: A-B 24: f2 -> f2 : B'=-A, [ B>=0 && 0>=1+A && 0>=1+A+B ], cost: -A-B 25: f2 -> f2 : B'=-1, [ B>=A && 0>=2+B ], cost: -1-B 26: f2 -> f2 : A'=B, [ B>=1+A && A>=0 ], cost: -A+B 27: f2 -> f2 : A'=-1, [ A+B>=0 && B>=1+A && 0>=2+A ], cost: -1-A Chained accelerated rules (with incoming rules): Start location: f0 0: f0 -> f2 : [ A>=1 && B>=1 ], cost: 1 1: f0 -> f2 : [ A>=1 && 0>=1+B ], cost: 1 2: f0 -> f2 : [ 0>=1+A && B>=1 ], cost: 1 3: f0 -> f2 : [ 0>=1+A && 0>=1+B ], cost: 1 28: f0 -> f2 : B'=1, [ A>=1 && A>=B && B>=2 ], cost: B 29: f0 -> f2 : B'=1-A, [ A>=1 && 0>=1+B && A+B>=2 ], cost: A+B 30: f0 -> f2 : A'=1, [ 0>=1+B && 1>=A+B && A>=1+B && A>=2 ], cost: A 31: f0 -> f2 : A'=B, [ 0>=1+A && 0>=1+B && A>=1+B ], cost: 1+A-B 32: f0 -> f2 : B'=-A, [ 0>=1+A && B>=1 && 0>=1+A+B ], cost: 1-A-B 33: f0 -> f2 : B'=-1, [ 0>=1+A && B>=A && 0>=2+B ], cost: -B 34: f0 -> f2 : A'=B, [ A>=1 && B>=1 && B>=1+A ], cost: 1-A+B 35: f0 -> f2 : A'=-1, [ B>=1 && A+B>=0 && B>=1+A && 0>=2+A ], cost: -A Removed unreachable locations (and leaf rules with constant cost): Start location: f0 28: f0 -> f2 : B'=1, [ A>=1 && A>=B && B>=2 ], cost: B 29: f0 -> f2 : B'=1-A, [ A>=1 && 0>=1+B && A+B>=2 ], cost: A+B 30: f0 -> f2 : A'=1, [ 0>=1+B && 1>=A+B && A>=1+B && A>=2 ], cost: A 31: f0 -> f2 : A'=B, [ 0>=1+A && 0>=1+B && A>=1+B ], cost: 1+A-B 32: f0 -> f2 : B'=-A, [ 0>=1+A && B>=1 && 0>=1+A+B ], cost: 1-A-B 33: f0 -> f2 : B'=-1, [ 0>=1+A && B>=A && 0>=2+B ], cost: -B 34: f0 -> f2 : A'=B, [ A>=1 && B>=1 && B>=1+A ], cost: 1-A+B 35: f0 -> f2 : A'=-1, [ B>=1 && A+B>=0 && B>=1+A && 0>=2+A ], cost: -A ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: f0 28: f0 -> f2 : B'=1, [ A>=1 && A>=B && B>=2 ], cost: B 29: f0 -> f2 : B'=1-A, [ A>=1 && 0>=1+B && A+B>=2 ], cost: A+B 30: f0 -> f2 : A'=1, [ 0>=1+B && 1>=A+B && A>=1+B && A>=2 ], cost: A 31: f0 -> f2 : A'=B, [ 0>=1+A && 0>=1+B && A>=1+B ], cost: 1+A-B 32: f0 -> f2 : B'=-A, [ 0>=1+A && B>=1 && 0>=1+A+B ], cost: 1-A-B 33: f0 -> f2 : B'=-1, [ 0>=1+A && B>=A && 0>=2+B ], cost: -B 34: f0 -> f2 : A'=B, [ A>=1 && B>=1 && B>=1+A ], cost: 1-A+B 35: f0 -> f2 : A'=-1, [ B>=1 && A+B>=0 && B>=1+A && 0>=2+A ], cost: -A Computing asymptotic complexity for rule 28 Simplified the guard: 28: f0 -> f2 : B'=1, [ A>=B && B>=2 ], cost: B Solved the limit problem by the following transformations: Created initial limit problem: -1+B (+/+!), 1+A-B (+/+!), B (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n,B==n} resulting limit problem: [solved] Solution: A / n B / n Resulting cost n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: n Rule cost: B Rule guard: [ A>=B && B>=2 ] WORST_CASE(Omega(n^1),?) ---------------------------------------- (4) BOUNDS(n^1, INF)